How to Work with Fractions — Operations & Simplifying

Core Fraction Concepts and Terminology

Before performing operations, it’s essential to understand key terms:

• Numerator: The top number (e.g., 3 in 3/4), representing the part.
• Denominator: The bottom number (e.g., 4 in 3/4), representing the whole.
• Proper Fraction: Numerator < denominator (e.g., 2/5).
• Improper Fraction: Numerator ≥ denominator (e.g., 7/3).
• Mixed Number: A whole number plus a proper fraction (e.g., 2 1/3).
• Equivalent Fractions: Different fractions representing the same value (e.g., 1/2 = 2/4 = 3/6).

1. Adding and Subtracting Fractions

The golden rule: fractions must have a common denominator to be added or subtracted.

Step-by-Step Process:

1. Find a common denominator: Use the Least Common Multiple (LCM) of the denominators.
2. Convert each fraction: Multiply numerator and denominator by the same number to get the common denominator.
3. Add/subtract numerators: Keep the common denominator.
4. Simplify: Reduce to lowest terms using the Greatest Common Divisor (GCD).

Example: 2/3 + 3/8

• LCM of 3 and 8 = 24
• Convert: (2×8)/(3×8) = 16/24, (3×3)/(8×3) = 9/24
• Add: 16/24 + 9/24 = 25/24
• Simplify: Already in lowest terms → 1 1/24 (as mixed number)

2. Multiplying Fractions

Multiplication is straightforward—no common denominator needed.

Step-by-Step Process:

1. Multiply numerators: Top × top.
2. Multiply denominators: Bottom × bottom.
3. Simplify: Reduce before or after multiplying.

Pro Tip: Cross-Canceling
Before multiplying, cancel common factors between any numerator and any denominator.

Example: 3/5 × 5/12

• Cancel 5: 3/1 × 1/12 = 3/12
• Simplify: 3/12 = 1/4

3. Dividing Fractions

Division is multiplication by the reciprocal (flip the second fraction).

Step-by-Step Process:

1. Keep the first fraction.
2. Change ÷ to ×.
3. Flip the second fraction (reciprocal).
4. Multiply and simplify.

Example: 5/6 ÷ 10/9

• Reciprocal of 10/9 = 9/10
• Multiply: 5/6 × 9/10
• Cross-cancel: 5 & 10 → 1 & 2; 9 & 6 → 3 & 2 → 1/2 × 3/2 = 3/4

4. Simplifying Fractions (Reducing to Lowest Terms)

A fraction is simplified when numerator and denominator share no common factors other than 1.

Method:

1. Find the GCD of numerator and denominator.
2. Divide both by the GCD.

Example: Simplify 18/24

• GCD of 18 and 24 = 6
• 18÷6 = 3, 24÷6 = 4 → 3/4

5. Converting Between Mixed Numbers and Improper Fractions

• Mixed → Improper: (Whole × Denominator) + Numerator / Denominator
  Example: 3 2/5 = (3×5 + 2)/5 = 17/5
• Improper → Mixed: Divide numerator by denominator. Quotient = whole, remainder = new numerator.
  Example: 17/5 = 3 R2 → 3 2/5

Pro Tips & Common Mistakes

• Always convert mixed numbers to improper fractions before operating—this avoids errors.
• Use cross-canceling in multiplication to keep numbers small.
• Never add denominators: 1/2 + 1/3 ≠ 2/5—this is the #1 mistake.
• Check your work: Convert answers back to decimals to verify (e.g., 3/4 = 0.75).
• For subtraction, borrow correctly: In mixed numbers, borrow 1 whole = denominator/denominator (e.g., 2 1/4 – 3/4 = 1 5/4 – 3/4 = 1 2/4 = 1 1/2).
• Simplify early: Reducing before final steps prevents large numbers.

Practical Applications

• Cooking: Doubling a recipe (e.g., 1 1/2 cups × 2 = 3 cups).
• Carpentry: Measuring cuts (e.g., 7 3/4" – 2 1/2" = 5 1/4").
• Finance: Calculating partial payments (e.g., 3/4 of a £100 bill = £75).
• Probability: Combining odds (e.g., 1/3 chance + 1/6 chance = 1/2).
• Education: Understanding test scores (e.g., 18/25 = 72%).